Difference between revisions of "2006 AMC 10B Problems/Problem 16"
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== Problem == | == Problem == | ||
− | Leap Day, February 29, 2004, | + | Leap Day, February 29, 2004, occurred on a Sunday. On what day of the week will Leap Day, February 29, 2020, occur? |
− | <math> \mathrm{(A) \ } Tuesday \qquad \mathrm{(B) \ } Wednesday \qquad \mathrm{(C) \ } Thursday \qquad \mathrm{(D) \ } Friday \qquad \mathrm{(E) \ } Saturday </math> | + | <math> \mathrm{(A) \ } \textrm{Tuesday} \qquad \mathrm{(B) \ } \textrm{Wednesday} \qquad \mathrm{(C) \ } \textrm{Thursday} \qquad \mathrm{(D) \ } \textrm{Friday} \qquad \mathrm{(E) \ } \textrm{Saturday} </math> |
== Solution == | == Solution == | ||
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<math> 16 \cdot 365 + 4 \cdot 1 = 5844 </math> | <math> 16 \cdot 365 + 4 \cdot 1 = 5844 </math> | ||
− | Since the days of the week repeat every <math>7</math> days and <math> 5844 \equiv -1 \bmod{7}</math>, the day of the week Leap Day 2020 occurs is the day of the week the day before Leap Day 2004 occurs which is <math>Saturday \Rightarrow E </math> | + | Since the days of the week repeat every <math>7</math> days and <math> 5844 \equiv -1 \bmod{7}</math>, the day of the week Leap Day 2020 occurs is the day of the week the day before Leap Day 2004 occurs, which is <math>\textrm{Saturday} \Rightarrow E </math>. |
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+ | == Solution 2 (Feasible Shortcuts) == | ||
+ | |||
+ | Since every non-leap year there is one day extra after the <math>52</math> weeks, we can deduce that if we were to travel forward <math>x</math> amount of non-leap years, then the answer would be "Sunday + <math>x</math>" days. | ||
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+ | However, we also have leap-years in our pool of years traveled forward, and for every leap-year that passes, we have two extra days after the 52 weeks. So we would travel "Sunday + <math>2x</math>" days. | ||
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+ | Mixing these two together, we get <math>4</math> leap years <math>(2008, 2012, 2016, 2020)</math>, and <math>12</math> "normal years". We thus get <math>12 + 2\cdot 4 = 20</math> "extra days" after Sunday. Since seven days are in a week, we can get <math>20\mod 7 = 6</math>, and six days after Sunday is <math>\textrm{Saturday} \Rightarrow E </math>. | ||
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+ | *Note how we did not require any complex or time-consuming computation, and thus it will save crucial time, especially during a test like the AMC 10. | ||
+ | |||
== See Also == | == See Also == | ||
− | + | {{AMC10 box|year=2006|ab=B|num-b=15|num-a=17}} | |
+ | |||
+ | [[Category:Introductory Number Theory Problems]] | ||
+ | {{MAA Notice}} |
Latest revision as of 17:26, 22 January 2020
Problem
Leap Day, February 29, 2004, occurred on a Sunday. On what day of the week will Leap Day, February 29, 2020, occur?
Solution
There are days in a year, plus extra day if there is a Leap Day, which occurs on years that are multiples of 4 (with a few exceptions that don't affect this problem).
Therefore, the number of days between Leap Day 2004 and Leap Day 2020 is:
Since the days of the week repeat every days and , the day of the week Leap Day 2020 occurs is the day of the week the day before Leap Day 2004 occurs, which is .
Solution 2 (Feasible Shortcuts)
Since every non-leap year there is one day extra after the weeks, we can deduce that if we were to travel forward amount of non-leap years, then the answer would be "Sunday + " days.
However, we also have leap-years in our pool of years traveled forward, and for every leap-year that passes, we have two extra days after the 52 weeks. So we would travel "Sunday + " days.
Mixing these two together, we get leap years , and "normal years". We thus get "extra days" after Sunday. Since seven days are in a week, we can get , and six days after Sunday is .
- Note how we did not require any complex or time-consuming computation, and thus it will save crucial time, especially during a test like the AMC 10.
See Also
2006 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 15 |
Followed by Problem 17 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
All AMC 10 Problems and Solutions |
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